Changing the coefficient ring of a module along a ring homomorphism can mean restriction of scalars, extension of scalars, or coextension of scalars. The underlying module may be retained with a smaller ring action, or replaced by a tensor or Hom construction with the new ring.
For a ring homomorphism and a left -module , the left -action on is . This construction is right adjoint to restriction of scalars.
For a ring homomorphism , tensoring an -module with the -bimodule gives an -module. This functor is left adjoint to restriction of scalars.
For a ring homomorphism , an -module becomes an -module by the displayed action. Its underlying abelian group stays the same.
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In algebra, the concept of **change of rings** involves the study of a ring homomorphism and how it allows us to transfer structures and properties from one ring to another. This is particularly relevant in areas like algebraic geometry, representation theory, and commutative algebra.